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# Multilayerity within multilayerity? On multilayer assortativity in social networks

Contributed paper at the 2015 Sunbelt social networks conference at Brighton, UK.

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### Multilayerity within multilayerity? On multilayer assortativity in social networks

1. 1. Multilayerity Within Multilayerity? Assortative Mixing of Multiple Partitions in Mulilayer Networks Moses A. Boudourides1 & Sergios T. Lenis2 Department of Mathematics University of Patras, Greece 1Moses.Boudourides@gmail.com 2sergioslenis@gmail.com 8 June 2015 Examples and Python code: http://mboudour.github.io/2015/06/09/Multilayerity-Within-Multilayerity%3F.html M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
2. 2. Graph Partitions Let G = (V , E) be a (simple undirected) graph with (ﬁnite) set of vectors V and (ﬁnite) set of edges E. The terms “graph” and “(social) network” are used here interchangeably. If S1, S2, . . . , Ss ⊂ V (for a positive integer s) are s nonempty (sub)sets of vertices of G, the (ﬁnite) family of subsets S = S(G) = {S1, S2, . . . , Ss}, forms a partition (or s–partition) in G, when, for all i, j = 1, 2, · · · , s, i = j, Si ∩ Sj = ∅ and V = s i=1 Si . S1, S2, . . . , Ss are called groups (of vertices) of partition S. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
3. 3. For every graph, there exists (at least one) partition of its vertices. Two trivial graph partitions are: the |V |–partition into singletons of vertices and Spoint = Spoint(G) = {{v}: v ∈ V }, the 1–partition into the whole vertex set Stotal = Stotal(G) = {V }. If G is a bipartite graph with (vertex) parts U and V , the bipartition of G is denoted as: Sbipartition = Sbipartition(G) = {U, V }. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
4. 4. Examples of Nontrivial Graph Partitions 1 Structural (endogenous) partitions: Connected components Communities Chromatic partitions (bipartitions, multipartitions) Degree partitions Time slicing of temporal networks Domination bipartitions (ego– and alter–nets) Core–Periphery bipartitions 2 Ad hoc (exogenous) partitions: Vertex attributes (or labels or colors) Layers (or levels) M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
5. 5. Multilayer Partitions and Multilayer Graphs A multilayer partition of a graph G is a partition L = L(G) = {L1, L2, . . . , L } into ≥ 2 groups of vertices L1, L2, . . . , L , which are called layers. A graph with a multilayer partition is called a multilayer graph. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
6. 6. Ordering Partitions Let P = {P1, P2, . . . , Pp} and Q = {Q1, Q2, . . . , Qq} be two (diﬀerent) partitions of vertices of graph G = (V , E). Partition P is called thicker than partition Q (and Q is called thinner than P), whenever, for every j = 1, 2, · · · , q, there exists a i = 1, 2, · · · , p such that Qj ⊂ Pi . Partitions P and Q are called (enumeratively) equivalent, whenever P is both thicker and thinner than partition Q. Obviously, partitions P and Q are equivalent, whenever |V | ≥ pq and, for all i = 1, 2, · · · , p and j = 1, 2, · · · , q, Pi ∩ Qj = ∅. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
7. 7. Self–Similarity of Partitions A partition P of vertices of graph G (such that |V | ≥ p2) is called (enumeratively) self–similar, whenever, for any i = 1, 2, · · · , p, there is a p–(sub)partition Pi = {Pi 1, Pi 2, . . . , Pi p} of the (induced) subgraph Pi . In this case, graph G is called (enumeratively) self–similar with respect to partition P. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
8. 8. Graph Partitions as Enumerative Attribute Assignments An assignment of enumerative attributes (or discrete attribute assignment) to vertices of graph G is a mapping A: V → {1, 2, · · · , α}, for some α ≤ |V |, through which the vertices of G are classiﬁed according to the values they take in the set {1, 2, · · · , α}. Every p–partition P = {P1, P2, . . . , Pp} of vertices of G corresponds to a p–assignment AP of enumerative attributes to vertices of G distributing them in the groups of partition P. Conversely, every assignment of enumerative attributes to vertices of G taking values in the (ﬁnite) set {1, 2, · · · , α} corresponds to a partition Pα = {Pα 1 , Pα 2 , . . . , Pα p } of the vertices of G such that, for any k = 1, 2, · · · , α, Pα k = {v ∈ V : A(v) = k} = A−1 (k). Thus, vertex partitions and enumerative vertex attribute assignments are coincident. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
9. 9. Let P be a partition of G into p groups of vertices and A be an assignment of α discrete attributes to vertices of G. Partition P is called compatible with attribute assignment A, whenever partition P is thinner than partition Pα, i.e., whenever, for every k = 1, 2, · · · , α, there exists (at least one) i = 1, 2, · · · , p such that Pi = A−1 (k). Apparently, partition P is compatible with attribute assignment A if anly if p ≥ α. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
10. 10. Assortativity of a Partition Let P = {P1, P2, . . . , Pp} be a vertex partition of graph G. Identifying P to a p–assignment of enumerative attributes to the vertices of G , one can deﬁne (cf. Mark Newman, 2003) the (normalized) enumerative attribute assortativity (or discrete assortativity) coeﬃcient of partition P as follows: rP = tr MP − ||M2 P|| 1 − ||M2 P|| , where MP is the p × p (normalized) mixing matrix of partition P. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
11. 11. In general, −1 ≤ rP ≤ 1, where rP = 0 signiﬁes that there is no assortative mixing of graph vertices with respect to their assignment to the p groups of partition P, i.e., graph G is conﬁgured as a perfectly mixed network (the random null model). rP = 1 signiﬁes that there is a perfect assortative mixing of graph vertices with respect to their assignment to the p groups of partition P, i.e., the connectivity pattern of these groups is perfectly homophilic. When rP atains a minimum value, which lies in general in the range [−1, 0), this signiﬁes that there is a perfect disassortative mixing of graph vertices with respect to their assignment to the p groups of partition P, i.e., the connectivity pattern of these groups is perfectly heterophilic. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
12. 12. Assortative Mixing Among Partitions Let P = {P1, P2, . . . , Pp} and Q = {Q1, Q2, . . . , Qq} be two (diﬀerent) partitions of vertices of graph G. Then, for any i = 1, 2, · · · , p, j = 1, 2, · · · , q, Pi ∩ Qj = ∅ and the intersection of P and Q P ∩ Q = {Pi ∩ Qj : i = 1, 2, · · · , p, j = 1, 2, · · · , q} is also a vertex partition of G. Notice that partition P ∩ Q is compatible with any one of the following three discrete attribute assignments on G: AP , in which, for any group Pi ∩ Qj of P ∩ Q, all vertices of Pi ∩ Qj are assigned a value in the set {1, 2, · · · , p}, AQ, in which, for any group Pi ∩ Qj of Q ∩ Q, all vertices of Pi ∩ Qj are assigned a value in the set {1, 2, · · · , q} and AP∩Q, in which, for any group Pi ∩ Qj of P ∩ Q, all vertices of Pi ∩ Qj are assigned a value in the set {1, 2, · · · , pq}. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
13. 13. Thus, by focusing on partition P ∩ Q and the discrete attribute assignment AP, one deﬁnes the discrete assortativity coeﬃcient of partition P with regards to partition Q as rPQ = rP∩Q and the mixing matrix of partition P with regards to partition Q as MPQ = MP∩Q. Similarly, by focusing on partition P ∩ Q and the discrete attribute assignment AQ, one deﬁnes the discrete assortativity coeﬃcient of partition Q with regards to partition P, rQP, and the mixing matrix of partition Q with regards to partition P, MQP. In general, rPQ = rQP. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
14. 14. Special Cases If Q = Spoint, then rPSpoint is the attribute assortativity coeﬃcient rP of graph G equipped with the vertex attributes corresponding to partition P, i.e., rPSpoint = rP. If P = Stotal, then, for any partition Q, rStotalQ = 1. If G is bipartite and P = Sbipartition, then, for any partition Q, rSbipartitionQ = −1. M.A. Boudourides & S.T. Lenis Assortative Mixing of Multiple Partitions in Mulilayer Networks
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Sep. 13, 2017

Contributed paper at the 2015 Sunbelt social networks conference at Brighton, UK.

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