Graphlets in Multiplex Networks

We develop graphlet analysis for multiplex networks and discuss how this analysis can be extended to multilayer and multilevel networks as well as to graphs with node and/or link categorical attributes. The analysis has been adapted for two typical examples of multiplexes: economic trade data represented as a 957-plex network and 75 social networks each represented as a 12-plex network. We show that wedges (open triads) occur more often in economic trade networks than in social networks, indicating the tendency of a country to produce/trade of a product in local structure of triads which are not closed. Moreover, our analysis provides evidence that the countries with small diversity tend to form correlated triangles. Wedges also appear in the social networks, however the dominant graphlets in social networks are triangles (closed triads). If a multiplex structure indicates a strong tie, the graphlet analysis provides another evidence for the concepts of strong/weak ties and structural holes. In contrast to Granovetter’s seminal work on the strength of weak ties, in which it has been documented that the wedges with only strong ties are absent, here we show that for the analyzed 75 social networks, the wedges with only strong ties are not only present but also significantly correlated.


Method Description
A multilayer or multiplex network in plain terms is a normal network with the addition of link categorical attributes. These attributes create a set of possible links allowed between nodes. To start, we can consider a multilayer network with d layers. In this case a link can be in one of the possible d layers. In addition a link can connect any two layers. In summation a link can be of d + d 2 types. Compared to this, a multiplex network contains d plexes. A link can be in one of the d plexes, or it can be in any number of plexes together. Meaning that in this case a link can be in one of 2 d − 1 types. In other words links belong to the set E t , which is defined as the power set of R = {1, . . . , D}, without the empty set, E t = P(R) \ ∅. Defined in this way multiplex networks will have an exponential number of different sub-orbits. For this reason we can apply some reduction(transformation) of the set E t . In the main text we further explore two possible reductions (plex-count,plex-count with distinct links inside orbits). In general the method remains the same with a change of the E t (link attribute set) used.
A similar consideration can be done for directed networks, however in this paper we focus only on the multiplex case. With the addition of these link states we expand the possible states a graphlet orbit can take. Each of these states we define as a graphlet sub-orbit. Now the goal is to enumerate all the graphlet sub-orbits. However to efficiently enumerate them, graph symmetry must be taken into consideration. A symmetry between several links exists when their attributes can be interchanged and the graphlet orbit remains the same. When no symmetry exists between links a cross product of the possible link attributes will exhaust all the possible configurations. Otherwise, if several links are symmetrical than taking all the combinations with replacement of the link attribute set, will give all the possible configurations. By using these two enumerations all the graphlet orbit forms can be distinctly identified. The composition of all the orbits for multiplex graphlets up to degree 4 are presented in Table 1.

Orbit Orbit Alias
Orbit class Size of orbit Table 1. Sub-orbit breakdown for graphlets up to order 4 The two possible enumerations are presented by pseudo code below. Each of them takes a list of links or link groups and enumerates them accordingly. By combining these two functions with E t -the link attribute set, we distinctly identify all the graphlet sub-orbits up to order 3 (shown in Algorithm 1). Then once we have enumerated all the separate orbits, we can count all the degree, wedge-path wedge-star and triangle occurrences for every node in a particular graph. This is presented in Algorithm 2. 1 1 The code for the analysis presented in this work is located at github.

2/21
Algorithm 1 Enumeration of all the sub-orbits up to N = 3 if AreNeighbors(i, j) then 38:

Synthetic Networks
In continuation basic description of the algorithms is provided. For each of these graph types, we then sample multiple graph instances of different sparsities and sizes. To create a multiplex network we than combine these random graphs so that each plex comes from a separate graph instance (the full sampling procedure is explained in the main text).
Watts-Strogatz -The aim of this model is to simulate a small world network. It starts with a connected n-size grid, and then a certain percentage of the grid links are interchanged, to simulate a network where the number of hops between any two nodes remains small (around 6 hops).
Erdosh Renyi -This graph model creates a graph by randomly adding edges with a probability p.
Barabashi-Albert -This model is initialized with a randomly connected network, and then in each iteration a new node is linked to an existing one. The existing node is chosen with a probability dependent on it's degree. In this way nodes with higher degrees are preferred compared to more isolated nodes.
Power-Law -These graphs are constructed in a similar manner as barabashi-albert graphs, except they use an additional triangle forming probability which randomly creates triangles instead of links.

Social Networks
In 2006 a baseline survey was conducted in a total of 75 villages. The survey consisted of a village questionnaire, a full census that collected data on all households in the villages, and a detailed follow-up survey fielded to a subsample of individuals. In the village questionnaire, data was collected regarding the village leadership, the presence of pre-existing NGOs and savings self-help groups, and various geographical features of the area (such as rivers, mountains, and roads). The household census gathered demographic information, GPS coordinates and data on a variety of amenities (such as roofing material, type of latrine, quality of access to electric power, etc.) for every household in each village. After the village and household modules were completed, a detailed individual survey was administered to a subsample of villagers. Respondents were randomly selected, and we stratified sampling by religion and geographic sub-location. The individual surveys were administered to eligible members and their spouses, yielding a sample of about 46% of all households per village. The individual questionnaire asked for information including age, sub-caste, education, language, native home, and occupation. Most importantly, these individual surveys also included a module that collected social network data along thirteen dimensions, including names of friends or relatives who visit the respondent's home, names of those friends or relatives the respondent visits, who the respondent goes to pray with (at a temple, church, or mosque), from whom the respondent would borrow money, to whom the respondent would lend money, from whom the respondent would borrow or to whom the respondent would lend material goods(kerosene, rice), from whom the respondent gets advice, and to whom the respondent gives advice. The resulting data are unusually rich, since the networks cover entire villages and 13 types of relationships between any two individuals, and since there are a large number of surveyed villages. They are publicly available on the project's web page 1 . Figure 1. Summary correlation matrices from economic trade networks. We calculate strong correlations (> 0.7) of up to 2000 individual product (2,3,4)-plex networks, coming from 100 product class pairs/triplets/quadruples. For each product class tuple we extract frequent strong correlations (appearing in >60% of networks). Finally we show the number of product class tuples where a strong correlation appears. a show the frequency of strong 2-plex correlations with the first plex reduction (plex count), c shows the frequency of strong 2-plex correlations with the second plex reduction (plex-count with distinct plexes in orbit), while b,d show the frequent strong correlations of product class triplets/quadruples (second reduction). The colored boxes separate correlations between different orbits (0,1,2 or 3).    . Summary correlation matrices from social interaction networks. We create social relation multiplex networks, where the plexes are social relations. We find strong correlations (> 0.7) in (2,3,4)-plex networks, coming from 66 or 100 social relation pairs/triplets/quadruples. For each social relation tuple we extract frequent strong correlations (appearing in >60% of villages). Finally we show the number of social relation tuples where a strong correlation appears. a shows the frequency of strong 2-plex correlations with the first plex reduction (plex count), c shows the frequency of strong 2-plex correlations with the second plex reduction (plex-count with distinct plexes in orbit), while b,d show the frequent strong correlations of social relation triplets/quadruples (second reduction). The colored boxes separate correlations between different orbits (0,1,2 or 3).

Full correlation table for Economic trade and Social relations networks
Here we present the full correlation tables (for correlations occurring more than 70% of the chosen networks) for the two real networks. We tested our method on 100 (or 66) randomly chosen 2-plexes/3-plexes/4-plexes. The procedure is thoroughly explained in the main text, with the added case of 3-plex/4-plex networks, chosen in a similar way as the 2-plexes (randomly selected product class triplets/quadruples or randomly chosen social relation triplets/quadruples). We use the second plex reduction (plex-count with distinct plexes inside orbits) in all cases.