Abstract
Understanding and controlling spreading processes in networks is an important topic with many diverse applications, including information dissemination, disease propagation and viral marketing. It is of crucial importance to identify which entities act as influential spreaders that can propagate information to a large portion of the network, in order to ensure efficient information diffusion, optimize available resources or even control the spreading. In this work, we capitalize on the properties of the Ktruss decomposition, a trianglebased extension of the core decomposition of graphs, to locate individual influential nodes. Our analysis on real networks indicates that the nodes belonging to the maximal Ktruss subgraph show better spreading behavior compared to previously used importance criteria, including node degree and kcore index, leading to faster and wider epidemic spreading. We further show that nodes belonging to such dense subgraphs, dominate the small set of nodes that achieve the optimal spreading in the network.
Introduction
Spreading processes in complex networks have gained great attention from the research community due to the plethora of applications that they occur, ranging from the spread of news and ideas to the diffusion of influence and social movements and from the outbreak of a disease to the promotion of commercial products. Being able to understand the underlying mechanisms that govern such processes is a crucial task with direct applications in various fields, including epidemiology, collective dynamics and viral marketing.
Typically, the interactions among individuals are responsible for the formation of information pathways in the network and to this extend, their position and topological properties have direct effect to the spreading phenomena occurring in the network. That way, a fundamental aspect on understanding and controlling the spreading dynamics is the identification of influential spreaders that can diffuse information to a large portion of the network. For example, in the case of virus propagation, such as influenza, the transmission of the disease mainly depends on the extend of contacts of the infected person to the susceptible population; thus, being able to locate and vaccinate individuals with good spreading properties can prevent from a potential outbreak of the disease, leading to efficient strategies of epidemic control. In a similar way, suppose that our goal is to promote an idea or a product in order to be adopted by a large fraction of individuals in the network. A key idea behind viral marketing is the wordofmouth effect^{1}; individuals that have already adopted the product, recommend it to their friends who in turn do the same to their own social circle, forming a cascade of recommendations^{2}. The basic question here is how to target a few initial individuals (e.g., by giving them free samples of the product or explaining them the idea), that can maximize the spread of influence in the network, leading to a successful promotion campaign.
The problem of identifying nodes with good spreading properties in networks, can be further split in two subtopics: (i) identification of individual influential nodes and (ii) identification of a group of nodes that, by acting all together, are able to maximize the total spread of influence. In this work, we focus on the problem of identifying single influential spreaders in networks. A straightforward approach towards finding effective spreading predictors, is to consider node centrality criteria and in particular the one of degree centrality. In fact, several studies have examined how the existence of heavytailed degree distribution in realworld networks^{3,4,5} is related to cascading effects concerning the robustness of such complex systems^{4,6,7,8}. Nevertheless, there exist cases where a node can have arbitrarily high degree, while its neighbors are not wellconnected, making degree a not very accurate predictor of the spreading properties. For example, this can occur when a high degree node is located to the periphery of the network. In fact, the spreading properties of a node are strongly related to the ones of its neighbors in the graph and thus, global centrality criteria seem to be more appropriate for this task.
Towards this direction, several approaches have been proposed in the related literature. Lu et al.^{9} proposed LeaderRank, a random walkbased algorithm similar to PageRank^{10} for identifying influential users in social networks. Later, Li et al.^{11} extended LeaderRank to properly detect influential nodes in weighted networks. Chen et al.^{12} proposed a semilocal centrality measure which serves as a tradeoff between degree and other computationally complex measures (betweenness and closeness centrality). Additionally, Chen et al.^{13} proposed ClusterRank, a local ranking method that takes into account the clustering coefficient of a node while in another approach^{14}, the diversity of the paths that emanate from a node was considered. The main idea was that the spreading ability of a node may be reduced if its propagation depends only on a few paths, while the rest ones lead to dead ends.
Of particular importance is the work by Kitsak et al.^{15}, which stressed out that highly connected nodes or those having high betweenness and closeness centralities, have little effect on the range of the spreading process. The main finding of their work was that, less connected but strategically placed nodes in the core of the network, are able to disseminate information to a larger part of the population. To quantify the coreperiphery structure of networks, they applied the kcore decomposition algorithm^{16,17,18}—a pruning process that removes nodes which do not satisfy a particular degreebased threshold. Their results indicated that nodes belonging to the maximal kcore subgraph are able to infect a larger portion of the network, compared to node degree or betweenness centrality, making the kcore number of a node a more accurate spreading predictor. Furthermore, extracting the kcore subgraph is a more efficient task compared to the heavy computation required by some centrality criteria (e.g., betweenness). Nevertheless, the resolution of kcore decomposition is quite coarse; depending on the structure of the network, many nodes will be assigned the same kcore number at the end of the process, even if their spreading capability differs from each other. Furthermore, building upon the good performance of the kcore decomposition, several extensions have been proposed^{19,20,21,22,23,24,25} (see Supplementary Note 6).
Our proposed approach moves on a similar axis as the one by Kitsak et al.^{15}; we argue that the topological properties of the nodes play a crucial role towards understanding their spreading capabilities. In particular, we consider that only a relatively small fraction of the nodes extracted by the kcore decomposition method corresponds to highly influential nodes. To that end, we propose the Ktruss decomposition of a graph^{26,27,28}, a trianglebased extension of the kcore decomposition, as a more accurate method to identify privileged spreaders. The algorithm is able to extract a more refined and even more dense subgraph of the initial graph–compared to the kcore decomposition–as the Ktruss is structurally more close to a clique. In fact, the Ktruss subgraph corresponds to the core of a kcore that filters out less important information. We perform experiments on large scale realworld networks, showing that the nodes belonging to the maximal Ktruss subgraph of the network show better spreading behavior under the SIR epidemic model–compared to previously used importance criteria–leading to faster and wider epidemic spreading. Furthermore, the extracted nodes dominate the small set of nodes that achieve the optimal spreading in the network.
Results
Let be an undirected graph with nodes and edges. In graph theory, the Ktruss subgraph of a graph G, is defined as the largest subgraph where all edges belong to at least triangles, i.e., cycle subgraphs of length three^{26,27}. Respectively, an edge has truss number if it belongs to but not to . Let denotes the set of nodes belonging to the maximal Ktruss subgraph of the graph. In this article, we argue that this set contains highly influential nodes with good spreading properties. It has been shown that the maximal kcore and Ktruss subgraphs (i.e., maximum values for k, K) overlap, with the latter being a subgraph of the former; the Ktruss subgraph represents the most connected part of the corresponding kcore, leading to a significant reduction of the set of nodes with respect to their structural properties and position within the graph (see Supplementary Note 3). Building upon the fact that the nodes belonging to the maximal kcore of the graph have good spreading properties^{15}, here we further refine this set of the most influential nodes, showing that the nodes belonging to set defined above perform even better, leading to faster and wider epidemic spreading.
We study realworld networks arising from online social networking and communication platforms (all datasets are publicly available^{29}). In particular, we investigate the following network datasets: (i) EMAILENRON and (ii) EMAILEUALL, two email communication networks; (iii) EPINIONS which is an online social network created from the product review website Epinions.com; (iv) WIKIVOTE, a network created by all the voting data between administrators of Wikipedia; (v) WIKITALK, created by the interaction data between Wikipedia users; (vi) SLASHDOT, which is created by the friendship relationships in the technology review website Slashdot.org. All datasets are considered undirected and unweighted; also, the largest connected component was used in the experiments. High level characteristics of the networks are shown in Table 1 (see Supplementary Note 1 for more details about the datasets).
Before presenting the results about the spreading properties, we examine the maximum level of the Ktruss decomposition, i.e., value , for the various graphs. As we can observe from Table 1, values vary from dataset to dataset, but compared to the values of the kcore decomposition, they tend to be much smaller. This is rather expected since the Ktruss decomposition relies on triangle participation, which is a more strict criterion compared to node degree. This last point is also a justification for the differences on the number of nodes belonging to the truss set and core set (i.e., the set of nodes belonging to the maximal kcore subgraph of the graph  see Methods for more details). Although these sets are overlapping, the one that corresponds to Ktruss has significantly smaller size compared to the maximal kcore subgraph. This was also one of the motivations of the proposed work; since the nodes of the maximal kcore subgraph perform well in information spreading, how to further refine this set by selecting a small subset that is characterized by even better spreading properties.
Evaluating the spreading performance
In the experimental results that follow, we are comparing the spreading performance of the nodes belonging to the set (truss method), to those belonging to the set (core method), i.e., the nodes belonging to the maximal kcore excluding those that belong to the maximal Ktruss of the graph–since is subset of , as discussed above. The core method constitutes the basic baseline approach, since it has been shown that outperforms other well known node importance criteria such as betweenness centrality^{15}. For completeness in the experimental evaluation, we also compare the spreading capabilities of the nodes that belong to the maximal Ktruss subgraph to those belonging to the set that contains the highest degree nodes in the graph (top degree method); we choose high degree nodes to achieve fair comparison between the different methods.
To study the spreading process and evaluate the performance of the nodes extracted by the Ktruss decomposition method, we apply the SIR epidemic model^{30,31}. Initially, we set one node to be in the infected state I. This node corresponds to our single spreader, that is chosen by the Ktruss decomposition method (in general, the initial node can be any node of the graph; the same procedure is also performed for the baseline methods). The rest of the nodes are assigned to the susceptible state S. At each time step, the infected nodes can infect their susceptible neighbors with probability β (i.e., infection rate). Furthermore, the nodes that have been previously infected can recover from the disease with probability γ (i.e., recovery rate). The process is repeated until no more new nodes get infected. Let be the size of the population that is infected by the epidemic triggered by node v (average value over multiple executions of the model  see also Supplementary Note 4 for a more detailed description about the simulation of the spreading process). Setting high β values, a relatively large fraction of the nodes will be infected and thus the role of individual nodes in the spreading process is diminished. In our approach, we set β close to the epidemic threshold , where is the largest eigenvalue of the adjacency matrix of the network^{32}. We also set parameter , as used by Kitsak et al.^{15}. As we will present later, we have performed experiments with several values of β and γ and the results are persistent concerning the comparison of the proposed method to other baselines.
To evaluate the spreading efficiency of the methods, we focus on the following quantities: (i) the number of nodes that become infected at each time step of the process and the corresponding cumulative one; (ii) the total number of infected nodes at the end of the epidemic; (iii) the time step where the epidemic fades out. For each node, we repeat the simulation 100 times (10 times for the WIKITALK graph due to its large size) and report the average behavior. In each case, we repeat the above for all the respective nodes and calculate the average behavior for the nodes of each set (truss method versus the two baselines core and top degree). The experimental results are shown in Table 2. The values of parameter β of the SIR model for each graph, are shown in Table 1. Table 2 shows the number of the newly infected nodes for some of the first ten time steps of the spreading process, which we consider as the outbreak of the epidemic (see Supplementary Table 1 for an extended version of this table including the number of infected nodes for all the first ten steps of the process; also Supplementary Table 2 shows the cumulative number of infected nodes per step). We also report the total number of nodes that were infected at the end of the process (Final step) and the time step where the epidemic dies out (Max step).
As we can observe, the truss method achieves significantly higher infection rate during the first steps of the epidemic. Furthermore, in almost all cases, the total number of infected nodes at the end of the process (Final step) is larger, while the fade out occurs earlier (Max step). Lastly, as we discussed above, the number of nodes in the truss set is much smaller compared to the set (Table 1). By refining significantly the set of influential nodes in truss set , the “weaker” spreaders of are left in core set , explaining the inferior behavior of the core method compared to top degree. Some small deviations from this behavior are observed in the SLASHDOT and WIKITALK graphs. In the SLASHDOT graph, the best performance is achieved by the top degree method, which from the very first steps is able to infect a larger amount of nodes. In the case of the WIKITALK graph, although the total number of infected nodes at the end (Final step) of the epidemic is almost the same for all methods, the proposed truss method performs quite effectively during the first steps of the process. In fact, it significantly outperforms both baseline methods achieving an increase of almost 23% on the cumulative number of infected nodes compared to both core and top degree methods, at the sixth step of the process.
We have also computed the cumulative difference of the number of infected nodes per step achieved by the methods. Let be the number of infected nodes at step t achieved by the truss method (similar for core and top degree). We define the cumulative difference for the truss and core methods at step t as
Similarly, we can define the same quantity for the truss vs. top degree methods. The results for the EMAILENRON, EPINIONS and WIKIVOTE graphs are shown in Fig. 1 (see Supplementary Fig. 1 for the results of the EMAILEUALL and WIKITALK graphs). For each graph, we have performed experiments for two values of parameter β and . We observe that the cumulative difference of the number of nodes that are being infected at every step is always larger between truss and core than between truss and top degree. Both differences increase during the outbreak of the epidemic until they stabilize to the number of nodes which is actually the final difference of the number of nodes that got infected (i.e., entered state I of the SIR model) during the epidemic process of the two compared methods. Clearly, as in almost all cases the differences are always above zero, one can conclude to the effectiveness of information diffusion when the spreading is triggered by the nodes that belong to the maximal Ktruss subgraph.
Comparison to the optimal spreading
Since we lack groundtruth information about the best spreaders in the network, to further study the performance of the proposed Ktruss decomposition method, we have examined the spreading achieved by each node of the graph. More precisely, we set each node at the infected state I and simulate the spreading capabilities of this node using the SIR model, as described earlier. Figure 2 depicts the distribution of the nodes with respect to the infection size M, for the EMAILENRON and WIKIVOTE graphs (parameter β of the SIR model was set to for this experiment). In both cases, the axes of the plot have been set to logarithmic scale. As we can observe, the distribution of the infection size M is skewed; only a small percentage of nodes are highly influential, while the majority of the nodes are able to infect only a small portion of the graph (small values of infection size M). Thus, our goal is to examine how the nodes detected by the Ktruss decomposition are distributed on this small subset of spreadingefficient nodes. Note that, similar observations have been made for the rest of the graphs described at Table 1.
To that end, we rank the nodes of the graph, according to the infection size . Let
be the node that achieves that highest infection size among all nodes in the graph, i.e., . In order to examine how the nodes detected by the Ktruss decomposition are distributed among the most efficient (optimal) spreaders, we consider a variable size window W over the ranked nodes and define to be the fraction of nodes of set that can be found within W as follows:
where T_{W} is the set of nodes that are located in the window W of size (in a similar way, we can define for the nodes of the maximal kcore subgraph). We are interested in examining how the quantities and behave with respect to the size of the window W.
Figure 3 depicts the distribution of the toptruss and topcore nodes, for various sizes of window W (i.e., fractions of the most efficient spreaders). As we can observe, for almost all datasets, reaches the maximum value (i.e., 100%) relatively early and for small window sizes, compared to . The maximum value of indicates that we have found all the nodes belonging to set in the window of fractional size W. An early and intense upward trend of the curve implies that a large fraction of the nodes belonging to the set of interest or , corresponds to nodes with the best spreading properties on the graph. For example, in the EMAILEUALL graph, the maximum of the nodes of set is reached in window , while in the case of set in window . Thus, the nodes detected by the Ktruss decomposition method (set are better distributed among the most efficient spreaders, compared to those located by the kcore decomposition (set . A slightly different behavior is observed in the WIKITALK and SLASHDOT graphs; in both graphs, the values of and are very close to each other for almost all choices of window W, indicating that both sets have almost the same overlap with the set of optimal spreaders. Nevertheless, as we have already presented in Table 2, for those two datasets the spreading performance of the truss nodes achieved during the first steps of the epidemic is much better.
Furthermore, we are interested to study the distribution of the nodes’ truss number with respect to window W. Similar to what described above, we consider a fraction of the best spreaders in the graph (as specified by W) and we examine the distribution of all truss numbers (and not only the maximum one) within it. Since nodes with high truss number are of particular importance here, we have considered groups of nodes as follows: (i) individual groups for each of the top five truss numbers, i.e., to . That way, the first group contains nodes with truss number equal to , the second group of nodes with truss number and so on. (ii) The rest of the groups concern truss numbers in the range to , grouping together five consecutive truss numbers each time. For example, the sixth group contains nodes with truss number in the range to . Note that, the last group may contain less than five truss numbers.
Figure 4 depicts the distribution of truss numbers for various values of window W. The colors on each bar correspond to the groups of truss number (darker colors for truss numbers closer to the maximum one). As we can observe in most of the datasets, for small values of window W, a large number of the nodes belong to the first group, i.e., their truss number is the maximum one. Since in most of the cases only a tiny fraction of the nodes of the graph belong to the very first groups (i.e., close to , even for small window sizes we also observe nodes from groups that correspond to smaller truss numbers. As the window W increases, i.e., deviate from the optimal spreading behavior, groups of smaller truss numbers start to evolve. From these results, it is evident that the truss number is related to the spreading capabilities of the nodes. Until now, we had only examined the effect of the nodes that belong to the maximal Ktruss subgraph. However, from this experiment we can conclude that, in general, nodes with high truss number tend to have good spreading properties–with the truss number being highly related to the spreading effect.
Impact of infection and recovery rate on the spreading process
We have also examined the impact of the infection and recovery rate of the SIR model on the epidemic spreading achieved by the proposed method (truss) and the two baseline methods (core and top degree). To that end, we have simulated the spreading process for various settings of parameters β and γ, examining the cumulative number of infected nodes per step of the process (see Supplementary Note 5 for more details about this experiment and Supplementary Fig. 4 for the results). We observed that, as the recovery probability γ decreases, the number of infected nodes increases both during the first time steps of the process, as well as at the end of the epidemic. This behavior is expected as high recovery rate γ implies that most of the nodes will move to the R state of the SIR model–thus being inactive in subsequent iterations of the model. Regarding the relative performance of the methods, we observed that it is not affected by the value of γ; the proposed truss method outperforms both baselines for all different settings of parameter γ. In the second case where we retain the recovery rate γ constant while the infection probability is increasing, we observed that the number of infected nodes increases. However, for higher values of β, the total number of infected nodes is almost the same for all methods. This behavior is rather expected; by increasing the infection rate, the importance of individual nodes in the epidemic process is reduced. For these values of β, the difference between the methods can be observed during the outbreak of the epidemic (i.e., first steps of the process), where the truss method performs qualitatively better.
Discussion
Understanding and controlling the mechanisms that govern spreading processes in complex networks is a fundamental task in various domains, including disease propagation and viral marketing. Central to these tasks is the problem of identification of influential nodes with good spreading properties, that are able to diffuse information to a large part of the network. It has been empirically observed that widely used node centrality criteria such as degree and betweenness, have drawbacks when applied to find nodes with good spreading properties; a node may have a large number of neighbors but if it is located to the periphery of the network, its spreading capability is reduced. Kitsak et al.^{15} applied the kcore decomposition method in order to locate centrally placed individuals with good spreading properties; their observations suggested that the identified nodes outperform previously used criteria with respect to the spreading effectiveness. However, the main drawback of the kcore decomposition is that its resolution is quite coarse. Depending on the structure of the network, many nodes will be assigned the same kcore number, even if their spreading capability differs from each other (see also the results presented in Table 1 regarding the number of nodes of the maximal kcore subgraph of several real networks).
The fact that a relatively large fraction of the nodes that are extracted by the kcore decomposition method corresponds to highly influential nodes, was the motivating force behind our approach. To deal with this issue, we have considered the Ktruss decomposition of a network–a trianglebased extension of the kcore structure. By setting a more strict criterion upon which nodes are assigned into layers of the graph, we have shown that the Ktruss decomposition can effectively reduce the number of candidate influential spreaders in the network, as it further refines the set of nodes belonging to the maximal kcore subgraph (recall that the maximal Ktruss is a subgraph of the maximal kcore). Using the SIR epidemic model, we have shown that such spreaders have the ability to influence a greater part of the network during the first steps of the process; also the total fraction of influenced nodes at the end of the epidemic is higher, compared to the performance of the rest nodes that belong to the maximal kcore subgraph and the top degree nodes of the network. Our experimental results also indicate that the Ktruss decomposition filters out the best spreaders of the kcore structure; the spreading effectiveness of the remaining nodes is weakened and those nodes show even worst behavior compared to the top degree ones (as indicated by the comparison of the core method to the top degree).
To further examine the spreading performance of the nodes located by the Ktruss decomposition method, we studied the spreading achieved by each node in the graph. After ranking the nodes of the network with respect to their spreading effectiveness, we observed that those belonging to the maximal Ktruss subgraph are distributed well among the optimal spreaders of the graph, presenting better behavior compared to the remaining nodes of the maximal kcore subgraph. Furthermore, we observed that the truss number in general, is closely related to the spreading effect. The nodes of the network are distributed among the optimal spreaders (after ranking) in a way that a relationship to truss number occurs.
An important issue about the Ktruss decomposition method, is the computational complexity; it can be proportional to , where is the number of edges of the graph, since it requires the computation of the number of triangles that each node participates to. This is actually the main weak point of this method, compared to the widely used kcore decomposition of linear time complexity . However, in this work, we are mainly interested in the nodes that belong to the maximal Ktruss subgraph. By taking into account the fact that a Ktruss subgraph is contained within a core subgraph, we can speedup the computation by firstly reducing the graph to its maximal core in linear time and then performing further refinements to extract the Ktruss subgraph^{26}.
It is worth noticing that most of the extensions presented for the kcore decompositionbased approach of Kitsak et al.^{15}, can also be applied to our method (see Supplementary Note 6 for a description of some of these methods). One such case concerns the identification of multiple initial nodes, as our method is designed to detect single influential spreaders (this is the case of the influence maximization problem, where we should locate multiple initial seed nodes that are able to maximize the total spread of influence)^{33,34,35,36}. The naive solution of choosing multiple nodes from the maximal Ktruss subgraph will not perform well, since those nodes are clustered together in the graph and share many common neighborhood nodes. Thus, as suggested in the related literature^{15}, a good strategy is to also consider the distance between them, as expressed by the number of hops needed to reach each other.
So far we have studied the effect of our method in real datasets by simulating spreading cascades. It is of great interest to also consider the identification of influential spreaders by following real information flow in social networks, as has been suggested by Pei et al.^{37}. Unfortunately, a lot of difficulties arise in such a case considering the lack of ground truth information that can actually represent the diffusion of a specific idea as is simulated by epidemic models. Additionally, the problem of how to consider a time frame to analyze the influence of every node of the network arises, which can alter the results depending on the setting chosen. Finally, in real information flow, some nodes are not found performing any activity, making the comparison of the methods even harder. We have tested our method at the Facebook dataset^{38} and the nodes found after performing Ktruss decomposition tend to be more effective in terms of spreading compared to those located by the kcore decomposition.
Methods
kcore decomposition
Let be an undirected graph with nodes and edges and let H be a subgraph of G, i.e., . Subgraph H is defined to be a kcore subgraph of G, denoted by , if it is a maximal connected subgraph in which all nodes have degree at least k. Then, each node has a core number , if it belongs to a kcore but not to a core. We denote as the set of nodes with the maximum core number (i.e., the nodes of the kcore subgraph of G that corresponds to the maximum value of k)^{16}. It is evident that if all the nodes of the graph have degree at least one, i.e., , then the 1core subgraph corresponds to the whole graph, i.e., . Furthermore, assuming that is the icore of G, then the kcore subgraphs are nested, i.e., .
Computing the kcore decomposition of a graph can be done through a simple process that is based on the following property: to extract the kcore subgraph, all nodes with degree less than k and their adjacent edges should be recursively deleted^{16}. That way, beginning with , the algorithm removes all the nodes (and the incident edges) with degree equal or less than k, until no such nodes have been remained in the graph. Also notice that, removing edges that are incident to a node may cause reductions to the degree of neighboring nodes; the degree of some nodes may become at most k and thus, they should also be removed at this step of the algorithm. When all remaining nodes have degree , k is increased by one and the process is repeated until no more remaining nodes are left in the graph. Since each node and edge is removed exactly once, the running time of the algorithm is ^{39}. Batagelj and Zaveršnik later proposed an algorithm for kcore decomposition^{17}.
Ktruss decomposition
The Ktruss decomposition extends the notion of kcore using triangles, i.e., cycle subgraphs of length 3^{26,27}. Let be an undirected graph. We define as a triangle a cycle subgraph of nodes . Additionally, the set of triangles of G is denoted by . The support of an edge is defined as and expresses the number of triangles that contain edge e. Then, the Ktruss, , denoted by , is defined as the largest subgraph of G, where every edge is contained in at least triangles within the subgraph, i.e., . Respectively, the truss number of an edge is defined as . Thus, if , then the edge belongs to but not to , i.e., but . We use to denote the maximum truss number of any edge . Since the definition of Ktruss is per edge, we define as truss number of a node , denoted by , the maximum truss number of its incident edges, i.e., , where is the set of neighborhood nodes of v. We denote as the set of nodes with maximum node truss number (in other words, this set contains the nodes of the maximal Ktruss subgraph). The Kclass of a graph is defined as . Then, the Ktruss decomposition is defined as the task of finding the Ktruss subgraphs of G, for all . That is, the Ktruss can be obtained by the union of all edges that have truss number at least K, i.e., .
The computation of the Ktruss subgraph, for a specific value of , follows similar methodological procedure as the one of kcore, where instead of the degree of a node, we examine the number of triangles that the node participates to: remove all edges if they do not participate to at least triangles, i.e., . The time complexity of the method is and the space complexity . However, as we described in the Discussion section, here we are mostly interested to extract the nodes belonging to the maximal Ktruss subgraph. This can be done effectively due to the fact that the maximal Ktruss is a subgraph of the maximal kcore of the graph^{26}.
The SIR spreading model
To determine the spreading effect of specific nodes in the network, we apply the SusceptibleInfectedRevovered (SIR) epidemic model^{30,31,40}. The model assumes a population of N individuals, divided on the following three states. Susceptible (S): the individual is not yet infected, thus being susceptible to the epidemic; Infected (I): the individual has been infected with the disease and it is capable of spreading the disease to the susceptible population; Recovered (R): after an individual has experienced the infectious period, it is considered as removed from the disease and it is not able to be infected again or to transmit the disease to others (immune to further infection or death).
Initially, all the nodes of the network are set at the susceptible state S, except from the one that we are interested to examine its spreading performance which is set at the infected state I. Then, at each time step t of the process, every node that is on the I state can infect its susceptible neighbors with probability β (called infection rate) and afterwards it can recover with probability γ (called recovery rate). Note that, a node cannot directly pass from state I to state R during the same time step t of the process.
Additional Information
How to cite this article: Malliaros, F. D. et al. Locating influential nodes in complex networks. Sci. Rep. 6, 19307; doi: 10.1038/srep19307 (2016).
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Acknowledgements
F.D.M. is a recipient of the Google Europe Fellowship in Graph Mining and this research is supported in part by this Google Fellowship. M.E.G.R. is funded by a DigiCosme Ph.D. Fellowship. M.E.G.R. is funded by a DigiCosme Ph.D. Fellowship and this research is supported in part by Labex DigiCosme.
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F.D.M. and M.E.G.R. contributed equally to this work. All authors conceived the study, performed the numerical experiments, analyzed the data and wrote the manuscript. All authors approved the final version of the manuscript.
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Malliaros, F., Rossi, ME. & Vazirgiannis, M. Locating influential nodes in complex networks. Sci Rep 6, 19307 (2016). https://doi.org/10.1038/srep19307
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