DOE OSTI2017
Social networks with signed edges (+/-) play an important role in an area of social network theory called structural balance. In these networks, edges represent relationships that are labeled as either friendly (+) or hostile (-). A signed social network is balanced only if all cycles of three or more nodes in the graph have an odd number of hostile edges. A fundamental property of a balanced network is that it can be cleanly divided into 2 factions, where all relationships within each faction are friendly, and all relationships between members of different factions are hostile. The more unbalanced a network is, the more edges will fail to adhere to this rule, making factions more ambiguous. Social theory suggests unbalanced networks should be unstable, a finding that has been supported by research on gangs, which shows that unbalanced relationships are associated with greater violence, possibly due to this increased ambiguity about factional allegiances (Nakamura et al). One way to estimate the imbalance in a network, if only edge relationships are known, is to assign nodes to factions that minimize the number of violations of the edge rule described above. This problem is known to be computationally NP-hard. However, Facchetti et al. have pointed out that it is equivalent to an Ising model with a Hamiltonian that effectively counts the number of edge rule violations. Therefore, finding the assignment of factions that minimizes energy of the equivalent Ising system yields an estimate of the imbalance in the network. Based on the Ising model equivalence of the signed-social network balance problem, we have used the D-Wave 2X quantum annealing computer to explore some aspects of signed social networks. Because connectivity in the D-Wave computer is limited to its particular native topology, arbitrary networks cannot be represented directly. Rather, they must be “embedded” using a technique in which multiple qubits are chained together with special weights to simulate a collection of nodes with the required connectivity. This limits the size of a fully connected network in the D-Wave to about 50 simulated nodes, using all of the approximately 1150 qubits in the machine. In order to keep within this limitation, while exploring a problem of potential social relevance, we constructed time series of historical network snapshots from Stanford’s Mapping Militants Project, where nodes represent militant organizations, and edges represent either alliances or rivalries between organizations. We constructed two series from different theaters – Iraq and Syria – spanning timelines from about 2000 to 2016, each with networks whose maximum size was in the 20-30 node range. Computationally, our experience suggests D-Wave technology is promising, providing fast, nearly constant scaling of computational effort in the main part of the calculation that relies on the quantum annealing cycle. However, the cost of embedding an arbitrary network of interest in the D-Wave native topology scales poorly. If the embedding cost can be amortized relative to the annealing cycle, it may be possible to gain a substantial advantage over classical computing methods, provided a large enough network can be accommodated by partitioning into subnetworks or some similar strategy. In terms of our application to networks of militant organizations, we found a rise in network imbalance in the Syrian theater that appears to correspond roughly with the entrance of the Islamic State into a milieu already populated with other groups, a phenomenon we plan to explore in more detail. In these very preliminary results, we also noticed that during at least one period where both the size and imbalance of the network increased substantially, the imbalance per edge seemed to remain fairly steady. This may suggest some adaptive behavior among the participating factions, which may also warrant further exploration.
97 MATHEMATICS AND COMPUTING↗